The answer to 'why is my supply line slow draining' is that a clogged or restricted supply line is the most common cause. This can happen due to built-up mineral deposits, debris, or other blockages in the line. Symptoms include slow water flow, low water pressure, and the line draining slowly.
To troubleshoot, start by checking the supply line for any visible clogs or obstructions. You can try using a plunger or small snake tool to clear the line. If that doesn't work, the issue may be deeper in the plumbing system and you'll likely need to call a professional plumber to diagnose and fix the problem.